9 If-Then Statements:
Relations Involving Addition, Subtraction, Multiplication, Division, and Equality
- This chapter does not provide a model activity for elementary students
- Discuss basic properties that underlie arithmetic and beginning algebra
- Together with previously discussed topics this chapter explores the basis for most of arithmetic and beginning algebra
Relating Subtraction to Addition and Division to Multiplication
Addition and Subtraction
- 9 boys and 12 girls, how many children in the class?
- 21 children, 9 are boys, how many girls are in the class?
- Both problems describe the same situation
- Different problems can be made from this information
- Different problems require different methods for solving
- Addition
- Subtraction
- Missing addend
- Demonstrate the relation between addition and subtraction
- Addition or subtraction can be thought of as positive numbers where a whole is composed of two parts for the set of whole numbers
- This is commonly thought of as the missing addend model for subtraction
- Addend + addend = sum for addition of whole numbers
- Addend + missing addend = sum for subtraction of whole numbers (Sum – addend = missing addend)
The Relation between Addition and Subtraction
- If a – b = c, then a = c + b
- If d + e = f, then d = f – e and e = f – d
- Might use this relation in several commonly encountered contexts
- Number facts
- Fact families
- Can make learning facts easier and more robust
- Solving word problems
- Open number sentences
- Provides for alternative solutions to problems
- Provides context for discussing the relation between addition and subtraction
Teacher Commentary 9.1
- 3rd grade students
- Some subtracted to find answer, but realized addition could also be used
- Wrote a conjecture to see if subtraction would always work for this type of problem
- Students encouraged to find big ideas in their conjectures
- Teacher provided necessary mathematical language to help students focus thinking
- Students came up with generalized statement for this conjecture using symbols
- If + = , then - =
Multiplication and Division
- Related in essentially as addition and subtraction
- Commonly referred to in division as the missing factor model
- Factor x factor = product for multiplication of whole numbers
- Factor x missing factor = product for division of whole numbers (Product factor = missing factor)
The Relation between Multiplication and Division
- If a b = c, then a = c x b
- If d x e = f, then d = f e for e 0, and e = f d for d 0
- Critical for learning division number facts
Operating on Both Sides of the Equal Sign
- 3rd grade students using relational thinking
- 345 + 576 = 342 + 574 + d
Operating on Both Sides of the Equal Sign
- If a = b, then a + c = b + c
- If a = b, then a x c = b x c
- If a = b, then a - c = b - c
- If a = b, then a c = b c, c 0
- Play a central role in solving algebraic equations
- See 9.1 p. 127
Proving Conjectures about Subtraction and Division
- Prove by relating to corresponding conjectures for addition and multiplication
- Prove d – d = 0 for all numbers
- Prove p 1 = p for all numbers
- Show why we cannot divide by zero: If r 0 = , then r = x 0. Thus r could only be zero, not any number
- Show why we cannot divide zero by zero: If 0 0 = , then 0 = x 0. Thus any number could replace , which contradicts the Fundamental Theorem of Arithmetic which says the prime factorization of a number is unique
- Can also consider fractions or the set of rational numbers for each of these ideas
Inverses
- For every number r except zero, there is a unique number such that
- Multiplicative inverseof r or the reciprocal of r
- Any division problem can be recast as a multiplication problem by multiplying by the inverse
- Any subtraction problem can be recast as an addition problem by adding the opposite
- Important for students to distinguish vocabulary between inverse and opposite
- Addition and multiplication have simpler rules than subtraction and division
- Does not mean we do away with subtraction and division, just want to understand the relation between them
The Basic Properties Revisited
- Some properties more critical than others
- Some properties derived from others
- See table 9.1 p. 130
- Properties of subtraction and division are missing – WHY?
- Zero property of multiplication not included – WHY?